3.03c Newton's second law: F=ma one dimension

1 A particle, of mass 3 kg , moves along a straight line. At time \(t\) seconds, the displacement, \(s\) metres, of the particle from the origin is given by $$s = 8 t ^ { 3 } + 15$$

1. Find the velocity of the particle at time \(t\).
2. Find the magnitude of the resultant force acting on the particle when \(t = 2\).

1

1. Sphere P , of mass 10 kg , and sphere Q , of mass 15 kg , move with their centres on a horizontal straight line and have no resistances to their motion. \(\mathrm { P } , \mathrm { Q }\) and the positive direction are shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-2_332_803_434_712} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} Initially, P has a velocity of \(- 1.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is acted on by a force of magnitude 13 N acting in the direction PQ . After \(T\) seconds, P has a velocity of \(4.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and has not reached Q .
   1. Calculate \(T\). The force of magnitude 13 N is removed. P is still travelling at \(4.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides directly with Q , which has a velocity of \(- 0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Suppose that P and Q coalesce in the collision to form a single object.
   2. Calculate their common velocity after the collision. Suppose instead that P and Q separate after the collision and that P has a velocity of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) afterwards.
   3. Calculate the velocity of Q after the collision and also the coefficient of restitution in the collision.
2. Fig. 1.2 shows a small ball projected at a speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) below the horizontal over smooth horizontal ground. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-2_424_832_1918_699} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure} The ball is initially 3.125 m above the ground. The coefficient of restitution between the ball and the ground is 0.6 . Calculate the angle with the horizontal of the ball's trajectory immediately after the second bounce on the ground.

4

1. A parachutist and her equipment have a combined mass of 80 kg . During a descent where the parachutist loses 1600 m in height, her speed reduces from \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and she does \(1.3 \times 10 ^ { 6 } \mathrm {~J}\) of work against resistances. Use an energy method to calculate the value of \(V\).
2. A vehicle of mass 800 kg is climbing a hill inclined at \(\theta\) to the horizontal, where \(\sin \theta = 0.1\). At one time the vehicle has a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is accelerating up the hill at \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) against a resistance of 1150 N .
   1. Show that the driving force on the vehicle is 2134 N and calculate its power at this time. The vehicle is pulling a sledge, of mass 300 kg , which is sliding up the hill. The sledge is attached to the vehicle by a light, rigid coupling parallel to the slope. The force in the coupling is 900 N .
   2. Assuming that the only resistance to the motion of the sledge is due to friction, calculate the coefficient of friction between the sledge and the ground.

1

1. In this part-question, all the objects move along the same straight line on a smooth horizontal plane. All their collisions are direct. The masses of the objects \(\mathrm { P } , \mathrm { Q }\) and R and the initial velocities of P and Q (but not R ) are shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-2_177_1011_488_529} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} A force of 21 N acts on P for 2 seconds in the direction \(\mathrm { PQ } . \mathrm { P }\) does not reach Q in this time.
   1. Calculate the speed of P after the 2 seconds. The force of 21 N is removed after the 2 seconds. When P collides with Q they stick together (coalesce) to form an object S of mass 6 kg .
   2. Show that immediately after the collision S has a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards R . The collision between S and R is elastic with coefficient of restitution \(\frac { 1 } { 4 }\). After the collision, S has a velocity of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of its motion before the collision.
   3. Find the velocities of R before and after the collision. \section*{(b) In this part-question take \(\boldsymbol { g } = \mathbf { 1 0 }\).} A particle of mass 0.2 kg is projected vertically downwards with initial speed \(5 \mathrm {~ms} ^ { - 1 }\) and it travels 10 m before colliding with a fixed smooth plane. The plane is inclined at \(\alpha\) to the vertical where \(\tan \alpha = \frac { 3 } { 4 }\). Immediately after its collision with the plane, the particle has a speed of \(13 \mathrm {~ms} ^ { - 1 }\). This information is shown in Fig. 1.2. Air resistance is negligible. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-2_383_341_1795_854} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
      \end{figure}
   4. Calculate the angle between the direction of motion of the particle and the plane immediately after the collision. Calculate also the coefficient of restitution in the collision.
   5. Calculate the magnitude of the impulse of the plane on the particle.

3 A particle \(P\), of mass 2 kg , is initially at rest at a point \(O\) on a smooth horizontal surface. The particle moves along a straight line, \(O A\), under the action of a horizontal force. When the force has been acting for \(t\) seconds, it has magnitude \(( 4 t + 5 ) \mathrm { N }\).

1. Find the magnitude of the impulse exerted by the force on \(P\) between the times \(t = 0\) and \(t = 3\).
2. Find the speed of \(P\) when \(t = 3\).
3. The speed of \(P\) at \(A\) is \(37.5 \mathrm {~ms} ^ { - 1 }\). Find the time taken for the particle to reach \(A\).

3 A particle of mass 0.2 kg lies at rest on a smooth horizontal table. A horizontal force of magnitude \(F\) newtons acts on the particle in a constant direction for 0.1 seconds. At time \(t\) seconds, $$F = 5 \times 10 ^ { 3 } t ^ { 2 } , \quad 0 \leqslant t \leqslant 0.1$$ Find the value of \(t\) when the speed of the particle is \(2 \mathrm {~ms} ^ { - 1 }\).
(4 marks)

1 A stone, of mass 2 kg , is moving in a straight line on a smooth horizontal sheet of ice under the action of a single force which acts in the direction of motion. At time \(t\) seconds, the force has magnitude \(( 3 t + 1 )\) newtons, \(0 \leqslant t \leqslant 3\). When \(t = 0\), the stone has velocity \(1 \mathrm {~ms} ^ { - 1 }\).
When \(t = T\), the stone has velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(T\).
(6 marks)

3 A particle of mass 0.5 kg is moving in a straight line on a smooth horizontal surface.
The particle is then acted on by a horizontal force for 3 seconds. This force acts in the direction of motion of the particle and at time \(t\) seconds has magnitude \(( 3 t + 1 ) \mathrm { N }\). When \(t = 0\), the velocity of the particle is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the magnitude of the impulse of the force on the particle between the times \(t = 0\) and \(t = 3\).
2. Hence find the velocity of the particle when \(t = 3\).
3. Find the value of \(t\) when the velocity of the particle is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

4 A smooth cylinder of radius \(a \mathrm {~m}\) is fixed with its axis horizontal and \(O\) is the centre of a cross-section. Particle \(P\), of mass 0.4 kg , and particle \(Q\), of mass 0.6 kg , are connected by a light inextensible string of length \(\pi a \mathrm {~m}\). The string is held at rest with \(P\) and \(Q\) at opposite ends of the horizontal diameter of the crosssection through \(O\) (see Fig. 1). The string is released and \(Q\) begins to descend. When \(O P\) has rotated through \(\theta\) radians, with \(P\) remaining in contact with the cylinder, the speed of each particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see Fig. 2). \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Show that \(v ^ { 2 } = 3.92 a ( 3 \theta - 2 \sin \theta )\) and find an expression in terms of \(\theta\) for the normal force of the cylinder on \(P\) at this time.
2. Given that \(P\) leaves the surface of the cylinder when \(\theta = \alpha\), show that \(\sin \alpha = k \alpha\) where \(k\) is a constant to be found.

2 One end of a light elastic string, of natural length 0.6 m and modulus of elasticity 30 N , is attached to a fixed point \(O\). A particle \(P\) of weight 48 N is attached to the other end of the string. \(P\) is released from rest at a point \(d \mathrm {~m}\) vertically below \(O\). Subsequently \(P\) just reaches \(O\).

1. Find \(d\).
2. Find the magnitude and direction of the acceleration of \(P\) when it has travelled 1.3 m from its point of release.

A particle \(P\) of mass 1.5 kg moves from rest at the origin such that at time \(t\) seconds it is subject to a single force of magnitude \(( 4 t + 3 ) \mathrm { N }\) in the direction of the positive \(x\)-axis.

1. Find the magnitude of the impulse exerted by the force during the interval \(1 \leq t \leq 4\). Given that at time \(T\) seconds, \(P\) has a speed of \(22 \mathrm {~ms} ^ { - 1 }\),
2. find the value of \(T\) correct to 3 significant figures.
   

1 A small smooth ring of mass 0.5 kg is travelling round a smooth circular wire, with centre O and radius 0.8 m . The circle of wire is in a horizontal plane. The speed of the ring, \(v \mathrm {~ms} ^ { - 1 }\), at time \(t \mathrm {~s}\) after passing through a point A on the wire is given by \(\mathrm { v } = 0.2 \mathrm { t } ^ { 2 } + 0.4 \mathrm { t } + 0.1\).

1. Find the angular speed of the ring 5 seconds after it passes through A .
2. Find the distance the ring travels along the wire in the first second after passing through A . At time \(T\) s after the ring passes through A the magnitude of the force exerted on the ring by the wire is 6.4 N . You may assume that any forces acting on the ring other than the force exerted on the ring by the wire and gravity can be ignored.
3. 1. Determine the value of \(T\).
   2. Hence find the tangential acceleration of the ring at this time.

5 On a fairground ride, the centre of a horizontal circular frame is attached to the top of a vertical pole, OP . When the frame and pole rotate, OP remains vertical and the frame remains horizontal. Chairs of mass 10 kg are attached to the frame by means of chains of length 2.5 m . The chains are modelled as being both light and inextensible. A side view of the situation when the ride is stationary is shown in Fig. 5. A chain fixed to point A on the circular frame supports a chair. The distance OA is 2 m . \begin{figure}[h] \end{figure} A child of mass 40 kg sits in a chair and, after a short time, the ride is rotating at a steady angular speed of \(\omega\) radians per second, with the chain inclined at an angle of \(50 ^ { \circ }\) to the downward vertical. The motion of the child and chair is in a horizontal circle.

1. Draw a sketch showing the forces acting on the chair when the ride is moving at this angular speed.
2. - Determine the tension in the chain.
   On another occasion, a man of mass 90 kg sits in the chair; after a short time, the ride is rotating in a horizontal circle at a steady speed of \(\omega\) radians per second, with the chain inclined at the same angle of \(50 ^ { \circ }\) to the downward vertical.
3. Without any detailed calculations, explain how your answers to part (b) for the child would compare with those for the man.
4. Explain why the chain is modelled as light.
5. State two other modelling assumptions that were used in answering part (b).

7 A builder ties two identical buckets, \(P\) and \(Q\), to the ends of a light inextensible rope. He hangs the rope over a smooth beam so that the buckets hang in equilibrium, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-007_360_296_502_904} The buckets are each of mass 0.6 kg .

1. 1. State the magnitude of the tension in the rope.
   2. State the magnitude and direction of the force exerted on the beam by the rope.
2. The bucket \(Q\) is held at rest while a stone, of mass 0.2 kg , is placed inside it. The system is then released from rest and, in the subsequent motion, bucket \(Q\) moves vertically downwards with the stone inside.
   1. By forming an equation of motion for each bucket, show that the magnitude of the tension in the rope during the motion is 6.72 newtons, correct to three significant figures.
   2. State the magnitude of the force exerted on the beam by the rope while the motion takes place.

7 A builder ties two identical buckets, \(P\) and \(Q\), to the ends of a light inextensible rope. He hangs the rope over a smooth beam so that the buckets hang in equilibrium, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-6_360_296_502_904} The buckets are each of mass 0.6 kg .

1. 1. State the magnitude of the tension in the rope.
   2. State the magnitude and direction of the force exerted on the beam by the rope.
2. The bucket \(Q\) is held at rest while a stone, of mass 0.2 kg , is placed inside it. The system is then released from rest and, in the subsequent motion, bucket \(Q\) moves vertically downwards with the stone inside.
   1. By forming an equation of motion for each bucket, show that the magnitude of the tension in the rope during the motion is 6.72 newtons, correct to three significant figures.
   2. State the magnitude of the force exerted on the beam by the rope while the motion takes place.

4 A ball is released from rest at a height of 15 metres above ground level.

1. Find the speed of the ball when it hits the ground, assuming that no air resistance acts on the ball.
2. In fact, air resistance does act on the ball. Assume that the air resistance force has a constant magnitude of 0.9 newtons. The ball has a mass of 0.5 kg .
   1. Draw a diagram to show the forces acting on the ball, including the magnitudes of the forces acting.
   2. Show that the acceleration of the ball is \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   3. Find the speed at which the ball hits the ground.
   4. Explain why the assumption that the air resistance force is constant may not be valid.

5 The constant forces \(\mathbf { F } _ { 1 } = ( 8 \mathbf { i } + 12 \mathbf { j } )\) newtons and \(\mathbf { F } _ { 2 } = ( 4 \mathbf { i } - 4 \mathbf { j } )\) newtons act on a particle. No other forces act on the particle.

1. Find the resultant force acting on the particle.
2. Given that the mass of the particle is 4 kg , show that the acceleration of the particle is \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
3. At time \(t\) seconds, the velocity of the particle is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\).
   1. When \(t = 20 , \mathbf { v } = 40 \mathbf { i } + 32 \mathbf { j }\). Show that \(\mathbf { v } = - 20 \mathbf { i } - 8 \mathbf { j }\) when \(t = 0\).
   2. Write down an expression for \(\mathbf { v }\) at time \(t\).
   3. Find the times when the speed of the particle is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

6 A small train at an amusement park consists of an engine and two carriages connected to each other by light horizontal rods, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-4_190_1038_420_493} The engine has mass 2000 kg and each carriage has mass 500 kg . The train moves along a straight horizontal track. A resistance force of magnitude 400 newtons acts on the engine, and resistance forces of magnitude 300 newtons act on each carriage. The train is accelerating at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Draw a diagram to show the horizontal forces acting on Carriage 2.
2. Show that the magnitude of the force that the rod exerts on Carriage 2 is 550 newtons.
3. Find the magnitude of the force that the rod attached to the engine exerts on Carriage 1.
4. A forward driving force of magnitude \(P\) newtons acts on the engine. Find \(P\).

4 A car, of mass 1200 kg , is connected by a tow rope to a truck, of mass 2800 kg . The truck tows the car in a straight line along a horizontal road. Assume that the tow rope is horizontal. A horizontal driving force of magnitude 3000 N acts on the truck. A horizontal resistance force of magnitude 800 N acts on the car. The car and truck accelerate at \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{81f3753c-f148-44be-8b35-0a8e531016dd-3_177_1002_580_513}

1. Find the tension in the tow rope.
2. Show that the magnitude of the horizontal resistance force acting on the truck is 600 N .
3. In fact, the tow rope is not horizontal. Assume that the resistance forces and the driving force are unchanged. Is the tension in the tow rope greater or less than in part (a)? Explain why.

2 A particle, of mass 2 kg , is attached to one end of a light inextensible string. The other end is fixed to the point \(O\). The particle is set into motion, so that it describes a horizontal circle of radius 0.6 metres, with the string at an angle of \(30 ^ { \circ }\) to the vertical. The centre of the circle is vertically below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-003_346_340_1580_842}

1. Show that the tension in the string is 22.6 N , correct to three significant figures.
2. Find the speed of the particle.

4 A car has a maximum speed of \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is moving on a horizontal road. When the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it experiences a resistance force of magnitude \(30 v\) newtons.

1. Show that the maximum power of the car is 52920 W .
2. The car has mass 1200 kg . It travels, from rest, up a slope inclined at \(5 ^ { \circ }\) to the horizontal.
   1. Show that, when the car is travelling at its maximum speed \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\) up the slope, $$V ^ { 2 } + 392 \sin 5 ^ { \circ } V - 1764 = 0$$
   2. Hence find \(V\).

5 A car, of mass 1600 kg , is travelling along a straight horizontal road at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the driving force is removed. The car then freewheels and experiences a resistance force. The resistance force has magnitude \(40 v\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car after it has been freewheeling for \(t\) seconds. Find an expression for \(v\) in terms of \(t\).

6 A particle is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). The particle is set into motion, so that it describes a horizontal circle whose centre is vertically below \(O\). The angle between the string and the vertical is \(\theta\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-6_506_442_534_794}

1. The particle completes 40 revolutions every minute. Show that the angular speed of the particle is \(\frac { 4 \pi } { 3 }\) radians per second.
2. The radius of the circle is 0.2 metres. Find, in terms of \(\pi\), the magnitude of the acceleration of the particle.
3. The mass of the particle is \(m \mathrm {~kg}\) and the tension in the string is \(T\) newtons.
   1. Draw a diagram showing the forces acting on the particle.
   2. Explain why \(T \cos \theta = m g\).
   3. Find the value of \(\theta\), giving your answer to the nearest degree.

1 A particle moves under the action of a force, \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), of the particle is given by $$\mathbf { v } = \left( t ^ { 3 } - 15 t - 5 \right) \mathbf { i } + \left( 6 t - t ^ { 2 } \right) \mathbf { j }$$

1. Find an expression for the acceleration of the particle at time \(t\).
2. The mass of the particle is 4 kg .
   1. Show that, at time \(t\), $$\mathbf { F } = \left( 12 t ^ { 2 } - 60 \right) \mathbf { i } + ( 24 - 8 t ) \mathbf { j }$$
   2. Find the magnitude of \(\mathbf { F }\) when \(t = 2\).

13 A vehicle, of total mass 1200 kg , is travelling along a straight, horizontal road at a constant speed of \(13 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) This vehicle begins to accelerate at a constant rate.
After 40 metres it reaches a speed of \(17 \mathrm {~ms} ^ { - 1 }\) Find the resultant force acting on the vehicle during the period of acceleration.